Algebra & Number Theory

Algebraic de Rham theory for weakly holomorphic modular forms of level one

Francis Brown and Richard Hain

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We establish an Eichler–Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasiperiods of modular forms.

Article information

Algebra Number Theory, Volume 12, Number 3 (2018), 723-750.

Received: 3 August 2017
Revised: 22 December 2017
Accepted: 22 January 2018
First available in Project Euclid: 28 July 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 11F23: Relations with algebraic geometry and topology 11F25: Hecke-Petersson operators, differential operators (one variable) 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

weakly holomorphic modular form algebraic de Rham cohomology


Brown, Francis; Hain, Richard. Algebraic de Rham theory for weakly holomorphic modular forms of level one. Algebra Number Theory 12 (2018), no. 3, 723--750. doi:10.2140/ant.2018.12.723.

Export citation


  • G. Bol, “Invarianten linearer differentialgleichungen”, Abh. Math. Sem. Univ. Hamb. 16:3-4 (1949), 1–28.
  • K. Bringmann, P. Guerzhoy, Z. Kent, and K. Ono, “Eichler–Shimura theory for mock modular forms”, Math. Ann. 355:3 (2013), 1085–1121.
  • R. Bruggeman, Y. Choie, and N. Diamantis, “Holomorphic automorphic forms and cohomology”, preprint, 2014.
  • L. Candelori, “Harmonic weak Maass forms of integral weight: a geometric approach”, Math. Ann. 360:1-2 (2014), 489–517.
  • R. F. Coleman, “Classical and overconvergent modular forms”, Invent. Math. 124:1-3 (1996), 215–241.
  • P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer, 1970.
  • W. Duke and P. Jenkins, “On the zeros and coefficients of certain weakly holomorphic modular forms”, Pure Appl. Math. Q. 4:4 (2008), 1327–1340.
  • M. Eichler, “Eine Verallgemeinerung der Abelschen Integrale”, Math. Z. 67 (1957), 267–298.
  • P. Guerzhoy, “Hecke operators for weakly holomorphic modular forms and supersingular congruences”, Proc. Amer. Math. Soc. 136:9 (2008), 3051–3059.
  • R. Hain, “Notes on the universal elliptic KZB connection”, 2013. To appear in Pure Appl. Math. Q.
  • N. M. Katz, “$p$-adic properties of modular schemes and modular forms”, pp. 69–190 in Modular functions of one variable, III (Antwerp, 1972), edited by W. Kuyk and J.-P. Serre, Lecture Notes in Math. 350, Springer, 1973.
  • N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Math. Studies 108, Princeton Univ. Press, 1985.
  • M. Kazalicki and A. J. Scholl, “Modular forms, de Rham cohomology and congruences”, Trans. Amer. Math. Soc. 368:10 (2016), 7097–7117.
  • Y. I. Manin, “Periods of parabolic forms and $p$-adic Hecke series”, Mat. Sb. $($N.S.$)$ 92(134):3 (1973), 378–401. In Russian; translated in Math. USSR-Sb. 21:3 (1973), 371–393.
  • A. J. Scholl, “Modular forms and de Rham cohomology; Atkin–Swinnerton–Dyer congruences”, Invent. Math. 79:1 (1985), 49–77.
  • G. Shimura, “Sur les intégrales attachées aux formes automorphes”, J. Math. Soc. Japan 11 (1959), 291–311.
  • H. P. F. Swinnerton-Dyer, “On $l$-adic representations and congruences for coefficients of modular forms”, pp. 1–55 in Modular functions of one variable, III (Antwerp, 1972), edited by W. Kuyk and J.-P. Serre, Lecture Notes in Math. 350, Springer, 1973. Correction in Modular functions of one variable, IV, Lecture Notes in Math. 476 (1975), 149.
  • D. Zagier, “Elliptic modular forms and their applications”, pp. 1–103 in The 1-2-3 of modular forms, edited by K. Ranestad, Springer, 2008.